Recursive formulas for arithmetic sequences | Mathematics I | High School Math | Khan Academy

G is a function that describes an arithmetic sequence. Here are the first few terms of the sequence: the first term is four, the second term is three and four-fifths, the third term is three and three-fifths, and the fourth term is three and two-fifths.

Find the values of the missing parameters a and b in the following recursive definition of the sequence. So they say the nth term is going to be equal to a if n is equal to 1, and it's going to be equal to g of n minus 1 plus b if n is greater than 1.

I encourage you to pause this video and see if you can figure out what a and b are going to be.

Well, the first one to figure out a is actually pretty straightforward. If n is equal to one, the first term when n equals one is four, so a is equal to four. We could write this as g of n is equal to 4 if n is equal to 1.

Now let's think about the second line. The second line is interesting. It's saying it's going to be equal to the previous term g of n minus 1. This means the n minus 1 term plus b will give you the nth term.

Let's just think about what's happening with this arithmetic sequence. When I go from the first term to the second term, what have I done? I have, it looks like I've subtracted one-fifth, so minus one-fifth. It's an arithmetic sequence, so I should subtract or add the same amount every time, and I am. I'm subtracting one-fifth.

So one way to think about it is, if we were to go the other way, we could say, for example, that g of 4 is equal to g of 3 minus one-fifth. You see that right over here? g of three is this; you subtract one-fifth, you get g of four.

Of course, I could have written this like g of 4 is equal to g of 4 minus 1 minus one-fifth. So when you look at it this way, you could see that if I'm trying to find the nth term, it's going to be the n minus 1 term plus negative one-fifth.

So b is negative one-fifth. Once again, if I'm trying to find the fourth term, if n is equal to four, I'm not going to use this first case because this has to be for n equals one.

If n equals four, I would use the second case, so it would be g of four minus one. It would be g of three minus one-fifth.

So we could say, g of n is equal to g of n minus one, the term right before that minus one-fifth if n is greater than one.

For the sake of this problem, we see that a is equal to four and b is equal to negative one-fifth.